ROME—Several hundred migrants are believed to have died in recent days as they attempted to sail to Italy, marking one of the largest losses of migrants’ lives at sea and highlighting the challenge Rome faces in coping with huge flows of refugees.

Refugees told aid workers Wednesday that more than 300 fellow passengers on four separate dinghies are likely to have died since the weekend as they attempted the treacherous Mediterranean crossing from Libya. The refugees were saved by Italian authorities, and on Wednesday...